Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $ x^{3} + 5 x + 57 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 14 a^{2} + 44 a + 50 + \left(29 a^{2} + 17 a + 23\right)\cdot 59 + \left(30 a^{2} + 12 a + 17\right)\cdot 59^{2} + \left(22 a^{2} + 15 a + 7\right)\cdot 59^{3} + \left(42 a^{2} + 29 a + 12\right)\cdot 59^{4} + \left(43 a^{2} + 27 a + 13\right)\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 34 a^{2} + 45 a + 38 + \left(a^{2} + 27 a + 49\right)\cdot 59 + \left(22 a^{2} + 24 a + 8\right)\cdot 59^{2} + \left(54 a^{2} + 43 a + 15\right)\cdot 59^{3} + \left(15 a^{2} + 32 a + 22\right)\cdot 59^{4} + \left(58 a^{2} + 33 a + 22\right)\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 11 a^{2} + 29 a + 40 + \left(28 a^{2} + 13 a\right)\cdot 59 + \left(6 a^{2} + 22 a + 16\right)\cdot 59^{2} + \left(41 a^{2} + 10\right)\cdot 59^{3} + \left(56 a + 50\right)\cdot 59^{4} + \left(16 a^{2} + 56 a + 38\right)\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 29 + 19\cdot 59 + 17\cdot 59^{2} + 22\cdot 59^{3} + 35\cdot 59^{4} + 50\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 57 + 39\cdot 59 + 15\cdot 59^{2} + 53\cdot 59^{3} + 39\cdot 59^{4} + 46\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 53 a^{2} + 7 a + 27 + \left(26 a^{2} + 7 a + 25\right)\cdot 59 + \left(36 a^{2} + 44 a + 37\right)\cdot 59^{2} + \left(9 a^{2} + 32 a + 15\right)\cdot 59^{3} + \left(15 a^{2} + 21 a + 56\right)\cdot 59^{4} + \left(13 a^{2} + 21 a + 45\right)\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 13 a^{2} + 35 a + 51 + \left(42 a^{2} + 52 a + 56\right)\cdot 59 + \left(16 a^{2} + 8 a + 10\right)\cdot 59^{2} + \left(28 a^{2} + 24 a + 58\right)\cdot 59^{3} + \left(21 a^{2} + 39 a + 37\right)\cdot 59^{4} + \left(23 a^{2} + 44 a + 20\right)\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 52 a^{2} + 17 a + 4 + \left(48 a^{2} + 58 a + 20\right)\cdot 59 + \left(5 a^{2} + 5 a + 53\right)\cdot 59^{2} + \left(21 a^{2} + 2 a + 53\right)\cdot 59^{3} + \left(22 a^{2} + 57 a + 40\right)\cdot 59^{4} + \left(22 a^{2} + 51 a + 56\right)\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(2,3)(6,7)$ |
| $(2,5)(4,7)$ |
| $(1,4)(2,7)(3,6)(5,8)$ |
| $(1,2)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-3$ |
| $3$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $-1$ |
| $3$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $1$ |
| $6$ | $2$ | $(1,2)(7,8)$ | $-1$ |
| $6$ | $2$ | $(1,4)(2,7)(3,6)(5,8)$ | $1$ |
| $8$ | $3$ | $(1,5,2)(4,7,8)$ | $0$ |
| $6$ | $4$ | $(1,6,2,4)(3,7,5,8)$ | $-1$ |
| $6$ | $4$ | $(1,5,2,3)(4,7,6,8)$ | $1$ |
| $8$ | $6$ | $(1,7,5,8,2,4)(3,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
